Elliptic and hyperelliptic functions describing the particle motion beneath small-amplitude water waves with constant vorticity

TL;DR

This paper derives analytic solutions for particle trajectories beneath small-amplitude water waves with constant vorticity, revealing non-closed paths expressed via elliptic and hyperelliptic functions, and discusses stagnation points.

Contribution

It introduces new analytic solutions involving elliptic and hyperelliptic functions for particle paths in vorticity-affected water waves, expanding understanding of fluid particle motion.

Findings

Particle paths are not closed curves.

Solutions expressed in elliptic and hyperelliptic functions.

Potential stagnation points due to vorticity.

Abstract

We provide analytic solutions of the nonlinear differential equation system describing the particle paths below small-amplitude periodic gravity waves travelling on a constant vorticity current. We show that these paths are not closed curves. Some solutions can be expressed in terms of Jacobi elliptic functions, others in terms of hyperelliptic functions. We obtain new kinds of particle paths. We make some remarks on the stagnation points which could appear in the fluid due to the vorticity.